Abstract

This thesis presents a unified framework for geometric discretization of highly oscillatory mechanics and classical field theories, based on Lagrangian variational principles and discrete differential forms. For highly oscillatory problems in mechanics, we present a variational approach to two families of geometric numerical integrators: implicit-explicit (IMEX) and trigonometric methods. Next, we show how discrete differential forms in spacetime can be used to derive a structure-preserving discretization of Maxwell's equations, with applications to computational electromagnetics. Finally, we sketch out some future directions in discrete gauge theory, providing foundations based on fiber bundles and Lie groupoids, as well as discussing applications to discrete Riemannian geometry and numerical general relativity.